A unified approach to mass transference principle and large intersection property

TL;DR

This paper unifies the mass transference principle and large intersection property in Diophantine approximation, providing a simpler, content-based approach that broadens their applicability and simplifies proofs.

Contribution

It establishes a general mass transference principle from Hausdorff content, unifying two key results and simplifying their proofs without constructing Cantor-like sets.

Findings

Unified proof for mass transference principle and large intersection property

Simpler verification process avoiding Cantor set constructions

Broader applicability of Hausdorff content-based methods

Abstract

The mass transference principle, discovered by Beresnevich and Velani [Ann Math (2), 2006], is a landmark result in Diophantine approximation that allows us to obtain the Hausdorff measure theory of $\limsup$ set. Another important tool is the notion of large intersection property, introduced and systematically studied by Falconer [J. Lond. Math. Soc. (2), 1994]. The former mainly focuses on passing between full (Lebesgue) measure and full Hausdorff measure statements, while the latter transfers full Hausdorff content statement to Hausdorff dimension. From this perspective, the proofs of the two results are quite similar but often treated in different ways. In this paper, we establish a general mass transference principle from the viewpoint of Hausdorff content, aiming to provide a unified proof for the aforementioned results. More precisely, this principle allows us to transfer the Hausdorff content bounds of a sequence of open sets $E_n$ to the full Hausdorff measure statement and large intersection property for $\limsup E_n$. One of the advantages of our approach is that the verification of the Hausdorff content bound does not require the construction of Cantor-like subset, resulting in a much simpler proof. As an application, we provide simpler proofs for several mass transference principles.